Use your brains -4

The first possible number which satisfies the given condition is 8. Go on adding LCM(3,5) i.e 15 to this number and you will get all the possible numbers like 23,38,53,68,83,98. Therefore the answer is 6....:)

$n$ is of the form $3a + 2$ and $5b + 3$ . The first value of $n = 8$ for which $a, b$ are integers. Adding the $LCM(3,5) = 15$ , we get all values of two-digit $n$

$n = \underbrace{\boxed{23, 38, 53, 68, 83, 98}}_{6 \text{ Values}}$

The numbers are 23, 53, 83, 18, 48, 78 i.e. 6 in total